The Price of Optimum in a Matching Game

  • Bruno Escoffier
  • Laurent Gourvès
  • Jérôme Monnot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6982)


Due to the lack of coordination, it is unlikely that the selfish players of a strategic game reach a socially good state. Using Stackelberg strategies is a popular way to improve the system’s performance. Stackelberg strategies consist of controlling the action of a fraction α of the players. However compelling an agent can be costly, unpopular or just hard to implement. It is then natural to ask for the least costly way to reach a desired state. This paper deals with a simple strategic game which has a high price of anarchy: the nodes of a simple graph are independent agents who try to form pairs. We analyse the optimization problem where the action of a minimum number of players shall be fixed and any possible equilibrium of the modified game must be a social optimum (a maximum matching).

For this problem, deciding whether a solution is feasible or not is not straitforward, but we prove that it can be done in polynomial time. In addition the problem is shown to be APX-hard, since its restriction to graphs admitting a vertex cover is equivalent, from the approximability point of view, to vertex cover in general graphs.


Nash Equilibrium Polynomial Time Vertex Cover Maximum Match Social Optimum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. The American Mathematical Monthly 69(1), 9–15 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Shapley, L.S., Shubik, M.: The assignment game i: The core. International Journal of Game Theory 1, 111–130 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kaporis, A.C., Spirakis, P.G.: Stackelberg games: The price of optimum. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms. Springer, Heidelberg (2008)Google Scholar
  4. 4.
    Korilis, Y.A., Lazar, A.A., Orda, A.: Achieving network optima using stackelberg routing strategies. IEEE/ACM Trans. Netw. 5(1), 161–173 (1997)CrossRefGoogle Scholar
  5. 5.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM 49(2), 236–259 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. Computer Science Review 3(2), 65–69 (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Papadimitriou, C.H.: Algorithms, games, and the internet. In: STOC, pp. 749–753 (2001)Google Scholar
  8. 8.
    Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. Theor. Comput. Sci. 410, 3327–3336 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Roughgarden, T.: Stackelberg scheduling strategies. SIAM J. Comput. 33(2), 332–350 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kumar, V.S.A., Marathe, M.V.: Improved results for stackelberg scheduling strategies. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 776–787. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Swamy, C.: The effectiveness of stackelberg strategies and tolls for network congestion games. In: SODA, pp. 1133–1142. SIAM, Philadelphia (2007)Google Scholar
  12. 12.
    Fotakis, D.: Stackelberg strategies for atomic congestion games. Theory Comput. Syst. 47(1), 218–249 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sharma, Y., Williamson, D.P.: Stackelberg thresholds in network routing games or the value of altruism. Games and Economic Behavior 67, 174–190 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bonifaci, V., Harks, T., Schäfer, G.: Stackelberg routing in arbitrary networks. Math. Oper. Res. 35(2), 330–346 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Manoussakis, Y.: Alternating paths in edge-colored complete graphs. Discrete Applied Mathematics 56(2-3), 297–309 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hochbaum, D.S.: Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston (1996)zbMATHGoogle Scholar
  17. 17.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Annals of Mathematics 162(1), 439–485 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Escoffier, B., Gourvès, L., Monnot, J.: Minimum regulation of uncoordinated matchings. CoRR abs/1012.3889 (2010)Google Scholar
  19. 19.
    Andelman, N., Feldman, M., Mansour, Y.: Strong price of anarchy. Games and Economic Behavior 65, 289–317 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Bruno Escoffier
    • 1
    • 2
  • Laurent Gourvès
    • 2
    • 1
  • Jérôme Monnot
    • 2
    • 1
  1. 1.LAMSADEUniversité de Paris-DauphineParisFrance
  2. 2.CNRS, UMR 7243ParisFrance

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